3.1 – Linear Equations and Inequalities
A linear relation compares two variables to see how they relate. The degree on both variables can
not exceed 1. As such, a linear relation is often called a first degree relation. In most cases this
relation will also be a function. A vertical line (i.e. x = ?) is the one exception.
Ex.
Ex.
As a formula
In function notation:
y=x+3
f(x) = x + 3
An equation sets a given relation or function, in most cases, equal to some value one is interested
in solving for. This is often called, “finding the zeros” because equation is re-arrange to this form
which is then where the graph will cross the x-axis (i.e. y = 0)
Ex.
Ex.
When is function f(x) = x + 3 equal to zero?
When is function f(x) = x + 3 equal to 5?
0=x+3
5=x+3
An inequality is interested in determining the domain (as an interval of values) when the function
satisfies the given scenario.
Ex.
Ex.
Example 1:
If p(h) = 5h then when is pay greater than $30?
If h(t) = 8t + 10 then when is height less then 18m?
Solve the following linear equations.
a) 2 x + 3x = 10
5x = 10
10
x=
One can always check
5
their answer by
x=2
b)
6 + 2 x = −4 − 3 x
2 x + 3 x = −4 − 6
5x = −10
−10
x=
5
x = −2
substituting back into
original equation.
Example 2:
Inequalities are
best read from
left to right. So
this says x is
greater than 3.
a)
5h > 30
h(t) < 18
8t + 10 < 18
Inequalities are best
set up exactly how
worded then rewritten afterwards
x
x
c) 2( x + 3) = 3(2 x − 2) d)
+4 = −2
2
3
2 x + 6 = 6x − 6
⎛x
⎞
⎛x
⎞
6⎜ + 4 ⎟ = 6⎜ − 2 ⎟
6 + 6 = 6x − 2 x
⎝2
⎠
⎝3
⎠
12 = 4 x
3 x + 24 = 2 x − 12
12
3 x − 2 x = −12 − 24
=x
4
x = −36
3= x
Solve the following linear inequalities and graph the solution.
3 x − 5 > 14
b)
3x > 9
x>3
x
− 3 ≤ −5
2
x − 6 ≤ −10
c) 4 x < 3 + 5 x
x ≤ −4
− 3 < 5x − 4x
−3< x
x is less than or
equal to -4.
A number line is a useful tool to think
about inequalities. Open circle tell one
not to include this value
3.1 – linear equations and inequalities
d)
Closed circle tell one that this
value is to be included.
Graph for c & d are the same
Keeping x
positive means
you have to say
-3 is less than x.
How does this
graph look?
4x < 3 + 5x
4 x − 5x < 3
−x<3
x > −3
A negative x means
you will switch the
inequality when
multiplying through
by -1. But this math
statement is easier to
say and think about?
3.1 – Linear Equations and Inequalities Practice Questions
1. Solve for unknown
a) 2x + 1 = 7
e) 2x = x – 9
i) 2x + 6 = 6x – 6
m) 8 + 3x = 4x
2. Solve for unknown
x x
a) + = 5
2 3
x +1
=3
e)
2
b) 5 – 3x = -7
f) 3x + 1 = 2x –7
j) –24 = 8(2y + 5)
n) –x = 4x - 6
2
y =1
3
2x + 1
= −5
f)
3
b) 4 +
c) 8x + 3 = 6
g) 15x + 2 = 12 – 10x
k) 2(x + 3) = 3(2x – 2)
o) 8 – 2x = 3x + 3
2
4
x−4 = −
3
5
x+3 x+5
=
g)
4
6
d) 2 = 8 – 2x
h) 3x – 2 = 1 – x
l) 2(y – 1) = - 4
p) 3y – 6 = 9(2 – y)
x
x
+1 = −1
3
2
2
3
h)
=
y −5 y + 2
c) −
d)
3. Write a statement to describe the following number line graphs.
a)
b)
c)
4. Solve the following inequalities.
a) 3x – 7 < 14
e) 2x + 6 ≤ 6x - 6
i) 8 + 3x ≤ 4x
2y + 3
≥ y+2
m)
3
b) 5x + 4x > 18
f) -3y ≥ -2y + 7
j) –x > 4x - 60
x x
n) + > 5
2 3
c) 2(y – 1) < - 4
g) 7x ≤ 16 – x
k) 8 – 2x < 3x + 13
2+ x 2
< x −1
o)
−5
3
d) 7x – 5x > 12
h) 4x ≤ 3x + 7
l) 3x ≤ 9(2 + x)
2
1
≥
p)
x − 1 3x
5. An employees pay is given by the function p(h) = 8h +10. Set up an inequality statement to
describe after how many hours the employees pay will exceed $74. Solve your inequality.
6. To raise money, the school basketball team is going to sell t-shirts. The cost of making the
shirts includes a fixed cost of $500 plus a cost of $7 per shirt printed. If the team intends to
sell the shirts for $15 each, what is the minimum number of shirts they need to sell to break
even?
Answers 1. a) 3 b) 4 c) 3/8 d) 3 e) -9 f) -8 g) 2/5 h) 3/4 i) 3 j) -4 k) 3 l) -1 m) 8 n) 6/5 o) 1 p) 2 2. a) 6 b) -4.5
c) -4.8 d) 12 e) 5 f) -8 g) 1 h) 19 3. a) x>3 b) x ≤ -2 c) -7 ≤ x< 3 4. a) x<7 b) x>2 c) y<-1d) x>6
e) 3 ≤ x or ≥ 3 f) y ≤ -7 g) x ≤ 2 h) x ≤ 7 i) 8 ≤ x j) x<12 k) -1<x l) x ≥ -3 m) y ≤ -3 n) x>6 o) x>9/13
p) x ≥ -1/5 5. 8h+10>74, h=8 6. 63 shirts
3.1 – linear equations and inequalities